**ECON 1095 QUANTITATIVE METHODS IN FINANCE**

* *

*Assignment 1 is due Sunday 2*^{nd}* September and contributes 25% to the assessment of this course.*

**INSTRUCTIONS**

*Please up load one (and one only) either word or pdf file. For the excel sections please just take screen shots of your work to show some of your workings then cut and paste into your document. For the maths sections, if you prefer you can hand write and scan and also add to word doc.*

**QUESTION 1**

**QUESTION 1**

Suppose you would like to model various financial processes and are trying to determine the appropriate mathematical functions to use. Please name and write down the general form of the equation for the most appropriate mathematical function in each of these situations.

- A time series process where the dependent variable changes by a constant absolute amount each period.
- A time series process where the dependent variable changes by a constant percentage amount each period.
- A situation where there is a constant elasticity relationship between the dependent and independent variables.
- For your answers in parts (b) and (c), write down the general forms of these functions using logarithms.

*(2 marks)*

*(2 marks)*

**QUESTION 2**

**QUESTION 2**

- Calculate both the average discrete and average continuous monthly returns for the following dividend adjusted share price and market index.

Share Price Market Index

Date Rit Rmt.

August-17 $1.55 1,175.00

September-17 $2.10 1,200.00

October-17 $2.45 1,305.00

November-17 $2.85 1,505.00

- Has this share under or over performed relative to the market?
- Suppose this share has a beta of 1.9 (b
_{i}= 1.9) and the annual risk free rate is 0.02. Would you invest?^{ }

*(1.5 marks)*

*(1.5 marks)*

**QUESTION 3**

**QUESTION 3**

For the following functions:

0 = 200×2 – 10x + 350

0 = 2×2 +20x + 5

0 = 6×2 + 5x – 20

- What are their discriminates?
- What is the significance of these discriminates?
- Complete the square and thereby solve for x for these equations (please do this both by hand and using the excel solver, excel instructions are on the following page).
- Use excel to graph these functions.

To use the solver (part (c)):

construct the formula for the equation 0 = 200×2 – 10x + 350; in B1 SUM(C1: E1), C1

=200*C2^2, D1 = -10*C2, E1 = +350, C2 the starting value for x.

go to solver (tools or formula), Set Cell $B$1, Equal To 0, By Changing C2, Solve.

keep trying different starting values of x until you have found all solutions.

if you have trouble go to help.

*(2 marks)*

*(2 marks)*

**QUESTION 4***(Use Excel to do this question)*

In the Excel file ECON1095 Data Sem 2 2018.xlsx on Canvas you will find the daily share prices for the top 200 ASX listed stocks, the market index and some government bond data for 2016, 2017 and 2018.

- Using the raw share prices and the market index calculate the continuous daily returns for the most recent 12 month period (6 July 2017 to 6 July 2018) and then find the
__average__continuous returns as well as the variances and standard deviations. Convert the average daily return values into annual returns (these annual average returns are all that is required for your answer). - Using the same sample period as in part (a), present three graphs showing the continuous returns for the market index against the highest returning, the lowest and the riskiest share.
- Comment on your graphs and briefly explain the limitations of using raw share prices.

*(3 marks)*

*(3 marks)*

**QUESTION 5**

**QUESTION 5**

Find the derivative with respect to x for these functions:

(a) y = 10 + 5x + x^{-2}

(b)

(c) + e^{(-3x)} + ln(25x^{-2})

Find the definite integral of the following function for values of x from 5 to 10:

(d) , that is find dx

*(4 marks)*

*(4 marks)*

**QUESTION 6**

**QUESTION 6**

For the equation y = 2 + 3 + 4x^{2} + 5x^{3}

Find the equation for the linear approximation when x = 3.

Find the equation for the quadratic approximation, also when x = 3.

Using excel to graph of the original function and the linear and quadratic approximations (same graph).

*(1.5 marks)*

*(1.5 marks)*

**QUESTION 7**

**QUESTION 7**

A pension fund must pay $10,000 at the end of each of the next five years. Find the Present Value and Duration for this set of payments using a discounted rate of 0.06 or 6 %.

Verify your answer to part (a) using excel.

Suppose the fund invests in a portfolio of two and five year zero-coupon bonds, and needs to satisfy the conditions to immunize its bond portfolio. It uses this portfolio to make these five payments of $10,000. Find the proportion of the fund allocated to 5-year bonds.

*(3 marks)*

*(3 marks)*

**QUESTION 8** *(use excel for this question)*

**QUESTION 8**

*(use excel for this question)*Assume you have four different bonds:

- B1 – A two year bond with a nominal rate of 3.5% per annum
- B2 – A five year bond with a nominal rate of 4.5 % per annum
- B3 – A ten year bond with a nominal rate of 5.5 % per annum
- B4 – A twenty-five year bond with a nominal rate of 6.5 % per annum

All these bonds have six monthly coupons and a face value of $2,000. Calculate their present values, Macauly durations and convexities using a YTM of 6% (YTM = 0.06).

*(3 marks)*

**QUESTION 9** *(use excel for this question)*

**QUESTION 9**

*(use excel for this question)*Suppose a fund manager is committed to making payments of $20,000 every 6 months for the next 15 years (an annuity). The fund manager uses a discount rate of 0.06 or 6 % pa.

- What is the present value of these payments?
- To fund these payments the fund manager must invest in the four bonds described in Question 8. Assume that she is trying to minimize transaction costs; use the figures in Question 8 to
__write__the equations that would need to be satisfied to immunize the annuity described in this question. Note that the fund manager is concerned that the application of these conditions could result in them only holding one or two different types of bonds. As this is considered risky she introduces a diversification condition whereby she must hold a minimum of five of each of B1, B2, B3 and B4. Note; these conditions will need to be considered in your equations. - Using the methods described in the course notes on Mathematical Programming; that is, using the solver in excel, find the portfolio of bonds that the fund manager must invest in to immunize the portfolio. Although you need to apply the diversification conditions, there is no need to apply the second order condition of Convexity
_{payments}> Convexity_{receipts}. Therefore, all that is required is for the two streams of payments and receipts to have the same present value, their Macaulay Durations must be equal and for the diversification conditions need to be satisfied. Submit you answer and sensitivity reports and write a brief paragraph explaining how much of each bond the fund manager should buy.

*(5 marks)*

*(5 marks)*

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